Introduction to Quantum Field Theory

Cover
Half-title page
Title page
Copyright page
Dedication
Contents
Preface
Acknowledgments
Introduction
I Quantum Fields, General Formalism, and Tree Processes
	1 Review of Classical Field Theory: Lagrangians, Lorentz Group and its Representations, Noether Theorem
		1.1 What is and Why Do We Need Quantum Field Theory?
		1.2 Classical Mechanics
		1.3 Classical Field Theory
		1.4 Noether Theorem
		1.5 Fields and Lorentz Representations
		Further Reading
		Exercises
	2 Quantum Mechanics: Harmonic Oscillator and Quantum Mechanics in Terms of Path Integrals
		2.1 The Harmonic Oscillator and its Canonical Quantization
		2.2 The Feynman Path Integral in Quantum Mechanics in Phase Space
		2.3 Gaussian Integration
		2.4 Path Integral in Configuration Space
		2.5 Correlation Functions
		Further Reading
		Exercises
	3 Canonical Quantization of Scalar Fields
		3.1 Quantizing Scalar Fields: Kinematics
		3.2 Quantizing Scalar Fields: Dynamics and Time Evolution
		3.3 Discretization
		3.4 Fock Space and Normal Ordering for Bosons
			3.4.1 Fock Space
			3.4.2 Normal Ordering
			3.4.3 Bose–Einstein Statistics
		Further Reading
		Exercises
	4 Propagators for Free Scalar Fields
		4.1 Relativistic Invariant Canonical Quantization
		4.2 Canonical Quantization of the Complex Scalar Field
		4.3 Two-Point Functions and Propagators
		4.4 Propagators: Retarded and Feynman
			4.4.1 Klein–Gordon Propagators
			4.4.2 Retarded Propagator
			4.4.3 Feynman Propagator
		Further Reading
		Exercises
	5 Interaction Picture and Wick Theorem for λφ[sup(4)] in Operator Formalism
		5.1 Quantum Mechanics Pictures
			5.1.1 Schrödinger Picture (Usual)
			5.1.2 Heisenberg Picture
			5.1.3 Dirac (Interaction Picture)
		5.2 Physical Scattering Set-up and Interaction Picture
			5.2.1 λφ[sup(4)] Theory
		5.3 Evolution Operator and the Feynman Theorem
		5.4 Wick's Theorem
		Further Reading
		Exercises
	6 Feynman Rules for λφ[sup(4)] from the Operator Formalism
		6.1 Diagrammatic Representation of Free Four-Point Function
		6.2 Interacting Four-Point Function: First-Order Result and its Diagrammatic Representation
		6.3 Other Contractions and Diagrams
		6.4 x-Space Feynman Rules for λφ[sup(4)]
		6.5 p-Space Feynman Rules and Vacuum Bubbles
			6.5.1 Canceling of the Vacuum Bubbles in Numerator vs. Denominator in Feynman Theorem
		Further Reading
		Exercises
	7 The Driven (Forced) Harmonic Oscillator
		7.1 Set-up
		7.2 Sloppy Treatment
		7.3 Correct Treatment: Harmonic Phase Space
		Further Reading
		Exercises
	8 Euclidean Formulation and Finite-Temperature Field Theory
		8.1 Phase-Space and Configuration-Space Path Integrals and Boundary Conditions
		8.2 Wick Rotation to Euclidean Time and Connection with Statistical Mechanics Partition Function
		8.3 Quantum-Mechanical Statistical Partition Function and Correlation Functions
		8.4 Example: Driven Harmonic Oscillator
		Further Reading
		Exercises
	9 The Feynman Path Integral for a Scalar Field
		9.1 Euclidean Formulation
		9.2 Perturbation Theory
		9.3 Dyson Formula for Perturbation Theory
		9.4 Solution of the Free Field Theory
		9.5 Wick's Theorem
		Further Reading
		Exercises
	10 Wick's Theorem for Path Integrals and Feynman Rules Part I
		10.1 Examples
		10.2 Wick's Theorem: Second Form
		10.3 Feynman Rules in x Space
		Further Reading
		Exercises
	11 Feynman Rules in x Space and p Space
		11.1 Proof of the Feynman Rules
		11.2 Statistical Weight Factor (Symmetry Factor)
		11.3 Feynman Rules in p Space
		11.4 Most General Bosonic Field Theory
		Further Reading
		Exercises
	12 Quantization of the Dirac Field and Fermionic Path Integral
		12.1 The Dirac Equation
		12.2 Weyl Spinors
		12.3 Solutions of the Free Dirac Equation
		12.4 Quantization of the Dirac Field
		12.5 The Fermionic Path Integral
			12.5.1 Definitions
		Further Reading
		Exercises
	13 Wick Theorem, Gaussian Integration, and Feynman Rules for Fermions
		13.1 Gaussian Integration for Fermions
			13.1.1 Gaussian Integration – The Real Case
			13.1.2 Real vs. Complex Integration
		13.2 The Fermionic Harmonic Oscillator and Generalization to Field Theory
		13.3 Wick Theorem for Fermions
		13.4 Feynman Rules for Yukawa Interaction
		Further Reading
		Exercises
	14 Spin Sums, Dirac Field Bilinears, and C, P, T Symmetries for Fermions
		14.1 Spin Sums
		14.2 Dirac Field Bilinears
		14.3 C, P, T Symmetries for Fermions
			14.3.1 Parity
			14.3.2 Time Reversal
			14.3.3 Charge Conjugation
		Further Reading
		Exercises
	15 Dirac Quantization of Constrained Systems
		15.1 Set-up and Hamiltonian Formalism
		15.2 System with Constraints in Hamiltonian Formalism: Primary/Secondary and First/Second-Class Constraints
		15.3 Quantization and Dirac Brackets
		15.4 Example: Electromagnetic Field
		Further Reading
		Exercises
	16 Quantization of Gauge Fields, their Path Integral, and the Photon Propagator
		16.1 Physical Gauge
		16.2 Quantization in Physical Gauge
		16.3 Lorenz Gauge (Covariant) Quantization
		16.4 Fadeev–Popov Path-Integral Quantization
		16.5 Photon Propagator
		Further Reading
		Exercises
	17 Generating Functional for Connected Green's Functions and the Effective Action (1PI Diagrams)
		17.1 Generating Functional of Connected Green's Functions
		17.2 Effective Action and 1PI Green's Functions
			17.2.1 Example: Free Scalar Field Theory in the Discretized Version
			17.2.2 1PI Green's Functions
		17.3 The Connected Two-Point Function
		17.4 Classical Action as Generating Functional of Tree Diagrams
		Further Reading
		Exercises
	18 Dyson–Schwinger Equations and Ward Identities
		18.1 Dyson–Schwinger Equations
			18.1.1 Specific Interaction
		18.2 Iterating the Dyson–Schwinger Equation
			18.2.1 Example
		18.3 Noether's Theorem
		18.4 Ward Identities
		Further Reading
		Exercises
	19 Cross-Sections and the S-Matrix
		19.1 Cross-Sections and Decay Rates
			19.1.1 Decay Rate
		19.2 In and Out States, the S-Matrix, and Wavefunctions
			19.2.1 Wavefunctions
		19.3 The Reduction Formula (Lehmann, Symanzik,Zimmermann)
		19.4 Cross-Sections from Amplitudes M
			19.4.1 Particle Decay
		Further Reading
		Exercises
	20 The S-Matrix and Feynman Diagrams
		20.1 Perturbation Theory for S-Matrices: Feynman and Wick
		20.2 Example: φ[sup(4)] Theory in Perturbation Theory and First-Order Differential Cross-Section
		20.3 Second-Order Perturbation Theory and Amputation
		20.4 Feynman Rules for S-Matrices
		Further Reading
		Exercises
	21 The Optical Theorem and the Cutting Rules
		21.1 The Optical Theorem: Formulation
		21.2 Unitarity: Optical Theorem at One Loop in λφ[sup(4)] Theory
		21.3 General Case and the Cutkovsky Cutting Rules
		Further Reading
		Exercises
	22*  Unitarity and the Largest Time Equation
		22.1 The Largest Time Equation for Scalars: Propagators
		22.2 Cut Diagrams
		22.3 The Largest Time Equation for Scalars: Derivation
		22.4 General Case
		Further Reading
		Exercises
	23 QED: Definition and Feynman Rules; Ward–Takahashi Identities
		23.1 QED: Definition
		23.2 QED Path Integral
		23.3 QED Feynman Rules
			23.3.1 Feynman Rules for Green's Functions in Euclidean Momentum Space
			23.3.2 Feynman Rules for S-Matrices in Minkowski Space
		23.4 Ward–Takahashi Identities
			23.4.1 Example 1: Photon Propagator
			23.4.2 Example 2: n-Photon Vertex Function for n ≥ 3
			23.4.3 Example 3: Original Ward–Takahashi Identity
		Further Reading
		Exercises
	24 Nonrelativistic Processes: Yukawa Potential, Coulomb Potential, and Rutherford Scattering
		24.1 Yukawa Potential
		24.2 Coulomb Potential
		24.3 Particle–Antiparticle Scattering
			24.3.1 Yukawa Potential
			24.3.2 Coulomb Potential
		24.4 Rutherford Scattering
		Further Reading
		Exercises
	25 e[sup(+)]e[sup(−)]  → l[bar(l)] Unpolarized Cross-Section
		25.1 e[sup(+)]e[sup(−)]  → l[bar(l)] Unpolarized Cross-Section: Set-up
		25.2 Gamma Matrix Identities
		25.3 Cross-Section for Unpolarized Scattering
		25.4 Center of Mass Frame Cross-Section
		Further Reading
		Exercises
	26 e[sup(+)]e[sup(−)]  → l[bar(l)] Polarized Cross-Section; Crossing Symmetry
		26.1 e[sup(+)]e[sup(−)]  → l[bar(l)] Polarized Cross-Section
		26.2 Crossing Symmetry
		26.3 Mandelstam Variables
		Further Reading
		Exercises
	27 (Unpolarized) Compton Scattering
		27.1 Compton Scattering: Set-up
		27.2 Photon Polarization Sums
		27.3 Cross-Section for Compton Scattering
		Further Reading
		Exercises
	28* The Helicity Spinor Formalism
		28.1 Helicity Spinor Formalism for Spin 1/2
		28.2 Helicity Spinor Formalism for Spin 1
		28.3 Amplitudes with External Spinors
		Further Reading
		Exercises
	29* Gluon Amplitudes, the Parke–Taylor Formula, and the BCFW Construction
		29.1 Amplitudes with External Gluons and Color-Ordered Amplitudes
		29.2 Amplitudes of Given Helicity and Parke–Taylor Formula
		29.3 Kleiss–Kluijf and BCJ Relations
		29.4 The BCFW Construction
		29.5 Application of BCFW: Proof of the Parke–Taylor Formula
		Further Reading
		Exercises
	30 Review of Path Integral and Operator Formalism and the Feynman Diagram Expansion
		30.1 Path Integrals, Partition Functions, and Green's Functions
			30.1.1 Path Integrals
			30.1.2 Scalar Field
		30.2 Canonical Quantization, Operator Formalism, and Propagators
		30.3 Wick Theorem, Dyson Formula, and Free Energy in Path-Integral Formalism
		30.4 Feynman Rules, Quantum Effective Action, and S-Matrix
			30.4.1 Feynman Rules in x Space (Euclidean)
			30.4.2 Simplified Rules
			30.4.3 Feynman Rules in p Space
			30.4.4 Simplified Momentum-Space Rules
			30.4.5 Classical Field
			30.4.6 Quantum Effective Action
			30.4.7 S-Matrix
			30.4.8 Reduction Formula (LSZ)
		30.5 Fermions
		30.6 Gauge Fields
		30.7 Quantum Electrodynamics
			30.7.1 QED S-Matrix Feynman Rules
		Further Reading
		Exercises
II Loops, Renormalization, Quantum Chromodynamics, and Special Topics
	31 One-Loop Determinants, Vacuum Energy, and Zeta Function Regularization
		31.1 Vacuum Energy and Casimir Force
		31.2 General Vacuum Energy and Regularization with Riemann Zeta ζ (−1)
		31.3 Zeta Function and Heat Kernel Regularization
			31.3.1 Heat Kernel Regularization
		31.4 Saddle Point Evaluation and One-Loop Determinants
			31.4.1 Path Integral Formulation
			31.4.2 Fermions
		Further Reading
		Exercises
	32 One-Loop Divergences for Scalars; Power Counting
		32.1 One-Loop UV and IR Divergences
		32.2 Analytical Continuation of Integrals with Poles
		32.3 Power Counting and UV Divergences
		32.4 Power-Counting Renormalizable Theories
			32.4.1 Examples
			32.4.2 Divergent φ[sup(4)] 1PI Diagrams in Various Dimensions
		Further Reading
		Exercises
	33 Regularization, Definitions: Cut-off, Pauli–Villars, Dimensional Regularization, and General Feynman Parametrization
		33.1 Cut-off Regularization and Regularizations of Infinite Sums
			33.1.1 Infinite Sums
		33.2 Pauli–Villars Regularization
		33.3 Derivative Regularization
		33.4 Dimensional Regularization
		33.5 Feynman Parametrization
			33.5.1 Feynman Parametrization with Two Propagators
			33.5.2 General One-Loop Integrals and Feynman Parametrization
			33.5.3 Alternative Version of the Feynman Parametrization
		33.6 Dimensionally Continuing Lagrangians
		Further Reading
		Exercises
	34 One-Loop Renormalization for Scalars and Counterterms in Dimensional Regularization
		34.1 Divergent Diagrams in φ[sup(4)] Theory in D=4 and its Divergences
			34.1.1 Divergent Parts
		34.2 Divergent Diagrams in φ[sup(3)] Theory in D=6 and its Divergences
			34.2.1 Divergent Parts
		34.3 Counterterms in φ[sup(4)] and φ[sup(3)] Theories
		34.4 Renormalization
			34.4.1 Examples
		Further Reading
		Exercises
	35 Renormalization Conditions and the Renormalization Group
		35.1 Renormalization of n-Point Functions
		35.2 Subtraction Schemes and Normalization Conditions
			35.2.1 Subtraction Schemes
			35.2.2 Normalization Conditions
		35.3 Renormalization Group Equations and Anomalous Dimensions
			35.3.1 Renormalization Group in MS Scheme
			35.3.2 φ[sup(4)] in Four Dimensions
		35.4 Beta Function and Running Coupling Constant
			35.4.1 Possible Behaviors for β(λ)
		35.5 Perturbative Beta Function in Dimensional Regularization in MS Scheme
			35.5.1 Examples
		35.6 Perturbative Calculation of γ[sub(m)] and γ[sub(d)] in Dimensional Regularization in the MS Scheme
		Further Reading
		Exercises
	36 One-Loop Renormalizability in QED
		36.1 QED Feynman Rules and Power-Counting Renormalizability
		36.2 Dimensional Regularization of Gamma Matrices
		36.3 Case 1: Photon Polarization  prod[sub(μν)](p)
		36.4 Case 2: Fermion Self-energy  sum(p)
		36.5 Case 3: Fermions–Photon Vertex  gamma[sub(μαβ)]
		Further Reading
		Exercises
	37 Physical Applications of One-Loop Results I: Vacuum Polarization
		37.1 Systematics of QED Renormalization
		37.2 Vacuum Polarization
		37.3 Pair Creation Rate
		Further Reading
		Exercises
	38 Physical Applications of One-Loop Results II: Anomalous Magnetic Moment and Lamb Shift
		38.1 Anomalous Magnetic Moment
		38.2 Lamb Shift
		Further Reading
		Exercises
	39 Two-Loop Example and Multiloop Generalization
		39.1 Types of Divergences at Two Loops and Higher
		39.2 Two Loops in φ[sup(4)] in Four Dimensions: Set-up
		39.3 One-Loop Renormalization
		39.4 Calculation of Two-Loop Divergences in φ[sup(4)] in Four Dimensions and their Renormalization
		Further Reading
		Exercises
	40 The LSZ Reduction Formula
		40.1 The LSZ Reduction Formula and Wavefunction Renormalization
		40.2 Adding Wavepackets
		40.3 Diagrammatic Interpretation
		Further Reading
		Exercises
	41* The Coleman–Weinberg Mechanism for One-Loop Potential
		41.1 One-Loop Effective Potential in λφ[sup(4)] Theory
		41.2 Renormalization and Coleman–Weinberg Mechanism
		41.3 Coleman–Weinberg Mechanism in Scalar-QED
		Further Reading
		Exercises
	42 Quantization of Gauge Theories I: Path Integrals and Fadeev–Popov
		42.1 Review of Yang–Mills Theory and its Coupling to Matter Fields
		42.2 Fadeev–Popov Procedure in Path Integrals
			42.2.1 Correlation Functions
		42.3 Ghost Action
		Further Reading
		Exercises
	43 Quantization of Gauge Theories II: Propagators and Feynman Rules
		43.1 Propagators and Effective Action
			43.1.1 Propagators
			43.1.2 Interactions
		43.2 Vertices
		43.3 Feynman Rules
		43.4 Example of Feynman Diagram Calculation
		Further Reading
		Exercises
	44 One-Loop Renormalizability of Gauge Theories
		44.1 Divergent Diagrams of Pure Gauge Theory
		44.2 Counterterms in MS Scheme
		44.3 Renormalization and Consistency Conditions
		44.4 Gauge Theory with Fermions
		Further Reading
		Exercises
	45 Asymptotic Freedom. BRST Symmetry
		45.1 Asymptotic Freedom
		45.2 BRST Symmetry
		45.3 Nilpotency of Q[sub(B)] and the Auxiliary Field Formulation
		Further Reading
		Exercises
	46 Lee–Zinn-Justin Identities and the Structure of Divergences (Formal Renormalization of Gauge Theories)
		46.1 Lee–Zinn-Justin Identities
		46.2 Structure of Divergences
		46.3 Solving the LZJ and Slavov–Taylor Identities
			46.3.1 Terms Linear in K[sup(a)][sub(μ)]
			46.3.2 Terms Linear in A and Not Containing K and L, and Linear in c
			46.3.3 Terms Quadratic in A and Not Containing K and L
			46.3.4 Terms Cubic in A and Not Containing K and L
		Further Reading
		Exercises
	47 BRST Quantization
		47.1 Review of the Dirac Formalism
			47.1.1 Dirac Brackets
		47.2 BRST Quantization
		47.3 Example of BRST Quantization: Electromagnetism in Lorenz Gauge
		47.4 General Formalism
			47.4.1 Quantum Action
		47.5 Example of General Formalism: Pure Yang–Mills
		47.6 Batalin–Vilkovisky Formalism (Field-Antifield)
		Further Reading
		Exercises
	48 QCD: Definition, Deep Inelastic Scattering
		48.1 QCD: Definition
		48.2 Deep Inelastic Scattering
			48.2.1 Parton Model
		48.3 Deep Inelastic Neutrino Scattering
		48.4 Normalization of the Parton Distribution Functions
		48.5 Hard Scattering Processes in Hadron Collisions
		Further Reading
		Exercises
	49 Parton Evolution and Altarelli–Parisi Equation
		49.1 QED Process
		49.2 Equivalent Photon Approximation
		49.3 Electron Distribution
		49.4 Multiple Splittings
			49.4.1 Boundary Conditions
			49.4.2 Photon Splitting into Pairs
		49.5 Evolution Equations for QED
		49.6 Altarelli–Parisi Equations and Parton Evolution
		Further Reading
		Exercises
	50 The Wilson Loop and the Makeenko–Migdal Loop Equation. Order Parameters; 't Hooft Loop
		50.1 Wilson Loop
			50.1.1 Abelian Case
			50.1.2 Nonabelian Case
		50.2 Wilson Loop and the Quark–Antiquark Potential
			50.2.1 Area Law and Perimeter Law
		50.3 The Makeenko–Migdal Loop Equation
			50.3.1 Path and Area Derivatives
			50.3.2 Makeenko–Migdal Loop Equation
		50.4 Order Parameters, 't Hooft Loop, Polyakov Loop
			50.4.1 't Hooft Loop
			50.4.2 Polyakov Loop
		Further Reading
		Exercises
	51 IR Divergences in QED
		51.1 Collinear and Soft IR Divergences
			51.1.1 Collinear Divergences
			51.1.2 Soft Divergences
			51.1.3 IR Divergences in Nonabelian Gauge Theories
		51.2 QED Vertex IR Divergence
		51.3 Dimensional Regularization Calculation
		51.4 Cancellation of IR Divergence by Photon Emission
		51.5 Summation of IR Divergences and Sudakov Factor
		Further Reading
		Exercises
	52 IR Safety and Renormalization in QCD: General IR-Factorized Form of Amplitudes
		52.1 QED Vertex: Eikonal Approximation, Exponentiation, and Factorization of IR Divergences
		52.2 IR Safety in QCD for Cross-Section for e[sup(+)]e[sup(-)] → Hadrons and Beta Function
			52.2.1 Born Cross-Section for e[sup(+)]e[sup(-)] → (q[bar(q)]) Hadrons
		52.3 Factorization and Exponentiation of IR Divergences in Gauge Theories
		Further Reading
		Exercises
	53 Factorization and the Kinoshita–Lee–Nauenberg Theorem
		53.1 The KLN Theorem
		53.2 Statement and Proof of Lemma
		53.3 Factorization and Evolution
			53.3.1 Factorization Theorem
		Further Reading
		Exercises
	54 Perturbative Anomalies: Chiral and Gauge
		54.1 Chiral Invariance in Classical and Quantum Theory
		54.2 Chiral Anomaly
			54.2.1 Anomaly in d=2 Euclidean Dimensions
			54.2.2 Anomaly in d=4 Dimensions
		54.3 Properties of the Anomaly
		54.4 Chiral Anomaly in Nonabelian Gauge Theories
		54.5 Gauge Anomalies
		Further Reading
		Exercises
	55 Anomalies in Path Integrals: The Fujikawa Method, Consistent vs. Covariant Anomalies, and Descent Equations
		55.1 Chiral Basis vs. V–A Basis
		55.2 Anomaly in the Path Integral: Fujikawa Method
		55.3 Consistent vs. Covariant Anomaly
		55.4 Descent Equations
		Further Reading
		Exercises
	56 Physical Applications of Anomalies, 't Hooft's UV–IR Anomaly Matching Conditions, and Anomaly Cancellation
		56.1 π[sup(0)] → γγ Decay
		56.2 Nonconservation of Baryon Number in Electroweak Theory
		56.3 The U(1) Problem
		56.4 't Hooft's UV–IR Anomaly Matching Conditions
		56.5 Anomaly Cancellation in General and in the Standard Model
			56.5.1 The Standard Model
		Further Reading
		Exercises
	57* The Froissart Unitarity Bound and the Heisenberg Model
		57.1 The S-Matrix Program, Analyticity, and Partial Wave Expansions
		57.2 The Froissart Unitarity Bound
			57.2.1 Application to Strong Interactions
		57.3 The Heisenberg Model for Saturation of the Froissart Bound
		Further Reading
		Exercises
	58 The Operator Product Expansion, Renormalization of Composite Operators, and Anomalous Dimension Matrices
		58.1 Renormalization of Composite Operators
		58.2 Anomalous Dimension Matrix
		58.3 Anomalous Dimension Calculation
			58.3.1 Tree Level: O(1)
			58.3.2 One-Loop Level: O(λ)
		58.4 The Operator Product Expansion
		58.5 QCD Example
		Further Reading
		Exercises
	59* Manipulating Loop Amplitudes: Passarino–Veltman Reduction and Generalized Unitarity Cut
		59.1 Passarino–Veltman Reduction of One-Loop Integrals
		59.2 Box Integrals
		59.3 Generalized Unitarity Cuts
		Further Reading
		Exercises
	60* Analyzing the Result for Amplitudes: Polylogs, Transcendentality, and Symbology
		60.1 Polylogs in Amplitudes
		60.2 Maximal and Uniform Transcendentality of Amplitudes
		60.3 Symbology
		Further Reading
		Exercises
	61* Representations and Symmetries for Loop Amplitudes: Amplitudes in Twistor Space, Dual Conformal Invariance, and Polytope Methods
		61.1 Twistor Space
		61.2 Amplitudes in Twistor Space
			61.2.1 Dual Space and Momentum Twistors
		61.3 Dual Conformal Invariance
		61.4 Polytopes and Amplitudes
		61.5 Leading Singularities of Amplitudes and a Conjecture for Them
		Further Reading
		Exercises
	62 The Wilsonian Effective Action, Effective Field Theory, and Applications
		62.1 The Wilsonian Effective Action
			62.1.1 φ[sup(4)] Theory in Euclidean Space
		62.2 Calculation of c[sub(delta)],[sub(i)]
		62.3 Effective Field Theory
			62.3.1 Nonrenormalizable Theories
			62.3.2 Removing the Cut-off
		Further Reading
		Exercises
	63 Kadanoff Blocking and the Renormalization Group: Connection with Condensed Matter
		63.1 Field Theories as Classical Spin Systems
		63.2 Kadanoff Blocking
		63.3 Expansion Near a Critical Point
		63.4 Critical Exponents (Near the Fixed Point)
		Further Reading
		Exercises
	64 Lattice Field Theory
		64.1 Continuum Limit
			64.1.1 Gaussian Fixed Point
		64.2 Beta Function
		64.3 Lattice Gauge Theory
		64.4 Lattice Gauge Theory: Continuum Limit
		64.5 Adding Matter
		Further Reading
		Exercises
	65 The Higgs mechanism
		65.1 Abelian Case
		65.2 Abelian Case: Unitary Gauge
		65.3 Abelian Case: Gauge Symmetry
		65.4 Nonabelian Case
			65.4.1 SU(2) Case
		65.5 Standard Model Higgs: Electroweak SU(2) × U(1)
		Further Reading
		Exercises
	66 Renormalization of Spontaneously Broken Gauge Theories I: The Goldstone Theorem and R[sub(ξ)] Gauges
		66.1 The Goldstone Theorem
		66.2 R[sub(ξ)] Gauges: Abelian Case
		66.3 R[sub(ξ)] Gauges: Nonabelian Case
		Further Reading
		Exercises
	67 Renormalization of Spontaneously Broken Gauge Theories II: The SU(2)-Higgs Model
		67.1 The SU(2)-Higgs Model
		67.2 Quantum Theory and LZJ Identities
		67.3 Renormalization
		Further Reading
		Exercises
	68 Pseudo-Goldstone Bosons, Nonlinear Sigma Model, and Chiral PerturbationTheory
		68.1 QCD, Chiral Symmetry Breaking, and Goldstone Theorem
		68.2 Pseudo-Goldstone Bosons, Chiral Perturbation Theory, and Nonlinear Sigma Model
		68.3 The SO(N) Vector Model
		68.4 Physical Processes and Generalizations
			68.4.1 Generalization
			68.4.2 Generalization to SU(3)
		68.5 Heavy Quark Effective Field Theory
		68.6 Coupling to Nucleons
		68.7 Mass Terms
		Further Reading
		Exercises
	69* The Background Field Method
		69.1 General Method and Quantum Partition Function
		69.2 Scalar Field Analysis for Effective Action
		69.3 Gauge Theory Analysis
		Further Reading
		Exercises
	70* Finite-Temperature Quantum Field Theory I: Nonrelativistic (''Manybody'') Case
		70.1 Review of Thermodynamics of Quantum Systems (Quantum Statistical Mechanics)
		70.2 Nonrelativistic QFT at Finite Temperature: ''Manybody'' Theory
		70.3 Paranthesis: Condensed Matter Calculations
		70.4 Free Green's Function
		70.5 Perturbation Theory and Dyson Equations
			70.5.1 Feynman Rules in x Space
			70.5.2 Feynman Rules in Momentum Space
			70.5.3 Dyson Equation
		70.6 Lehmann Representation and Dispersion Relations
		70.7 Real-Time Formalism
			70.7.1 Lehmann Representation and Dispersion Relations
			70.7.2 Relation with Green–Matsubara Functions
			70.7.3 Free Green–Zubarev Function
			70.7.4 Correlation Functions and Scattering
		Further Reading
		Exercises
	71* Finite-Temperature Quantum Field Theory II: Imaginary and Real-Time Formalisms
		71.1 The Imaginary-Time Formalism
		71.2 Imaginary-Time Formalism: Propagators
		71.3 KMS (Kubo–Martin–Schwinger) Relation
		71.4 Real-Time Formalism
		71.5 Interpretation of Green's Functions
		71.6 Propagators and Field Doubling
		Further Reading
		Exercises
	72* Finite-Temperature Quantum Field Theory III: Thermofield Dynamics and Schwinger–Keldysh ''In–In'' Formalism for Thermal and Nonequilibrium Situations. Applications
		72.1 Thermofield Dynamics
			72.1.1 Thermal Fermionic Harmonic Oscillator
			72.1.2 Bosonic Harmonic Oscillator
		72.2 The Schwinger–Keldysh Formalism at T=0
		72.3 Schwinger–Keldysh Formalism at Nonzero T
		72.4 Application of Thermal Field Theory: Finite-Temperature Effective Potential
		Further Reading
		Exercises
References
Index